Is X divisible by 3^k?

Question

Is X divisible by  3^k ?

(1) X = 30!
(2) X =  ^(k/3), where n and k/3 are positive integers

King407 · Answer

A)  30!/3^k , since k can assume any value, it will vary the result. --  Insufficient

B)  (3n+6)(3n+9)(n+1)]^\frac{k}{3}

This can be written as   ^\frac{k}{3}  =>  9^\frac{k}{3}* ^\frac{k}{3}

Now  X/3^k  =  9^\frac{k}{3}* ^\frac{k}{3}/3^k

=>  3^\frac{2k}{3}* ^\frac{k}{3}/3^k

=>  ^\frac{k}{3}/3^\frac{k}{3}

Since, (n+3)*(n+2)*(n+1) are consecutive integers, at least on of them is divisible by 3; and since numerator and denominator has same exponent, it is divisible ----  Sufficient

Hence Answer is B

Shree9975 · Answer

Bunuel  :- Can you please suggest or solve this by some easier method.
I didnt get what king did.

Thanks in advance

PrepTap · Answer

Hi Shree9975

Please let us know where exactly you are facing problem?

This question is about simplifying the algebraic equation.

Statement 1: As we do not know the value of k, we cannot say if 30! is divisible by 3^k or not.

Statement 2: X =  ^(k/3) =  ^(k/3) =  ^(k/3)

(n+1)(n+2)(n+3) is a product of 3 consecutive integers and as every third number is divisible by 3, a product of 3 consecutive integers is divisible by 3.

Hence, (n+1)(n+2)(n+3) = 3*p where p is an integer.

Thus X =  ^(k/3) =  ^(k/3) =  ^(k/3) = 3^k * p^(k/3)

So we can conclude that X is divisible by 3^k.

Answer: Option (B)

GMATinsight · Answer

Statement 1: X= 30!

The statement doesn't provide any information about k therefore this is insufficient to answer the Question

Statement 1: X =  ^(k/3), where n and k/3 are positive integers

i.e. X =  3 (n+2)x  3 (n+3)x (n+1)]^(k/3), where n and k/3 are positive integers

X =     ^(k/3)

but (n+2)x(n+3)x(n+1) is a product of three consecutive Integers and every three consecutive integer include one number as multiple of 3 therefore (n+2)x(n+3)x(n+1) will be a multiple of 3

Let (n+2)x(n+3)x(n+1) = 3a

X =     ^(k/3) =     (3a)^(k/3)

i.e. X =     (a)^(k/3)

i.e. X is divisible by 3^k, Hence  Sufficient

Answer: Option  B

Madhavi1990 · Answer

Went through a couple of answers, and I felt that I needed to further simplify the factorization.

St 2:  ^k/3 -->  ^k/3 => 9^k/3  /3^k

=> 3^2/3k / 3^k *  
=>3^k - 2/3k *   {BECAUSE larger power minus smaller power when base is the same}
=> 3^k  --> thus this expression is divisible by 3^k as not only the   is a multiple of 3, it is multiplied by the given denominator 3^k

bumpbot · Answer

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Is X divisible by 3^k?

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